Irreducible monomial linear groups of degree four over finite fields

We describe an algorithm for explicitly listing the irreducible monomial subgroups of GL(n, q), given a, suitable list of finite irreducible monomial subgroups of GL(n, ℂ), where n is 4 or a prime, and q is a prime power. Particular attention is paid to the case n = 4, and the algorithm is illustrated for n = 4 and q = 5. Certain primitive permutation groups can be constructed from a list of irreducible monomial subgroups of GL(n, q). The paper's final section shows that the computation of automorphisms of such permutation groups reduces mainly to computation of irreducible monomial subgroups of GL(n, q), q prime.